Standard Young Tableaux in the Weyl Group Setting

A generalization of the notion of standard Young tableau has recently arisen from work on the representation theory of affine Hecke algebras. In the generalized setting, a standard tableau is defined to be any element of a finite Weyl group whose inversion set satisfies a certain pair of intersection conditions. In this paper, we prove that the set of generalized standard tableaux of fixed shape, when nonempty, is a certain interval in the weak ordering. In addition, we establish a nonemptiness criterion for the set of standard tableaux of prescribed shape. These results are obtained for shapes that satisfy an integrality condition.